Definition

is called elliptic integral if R(t, s) is a rational
function of t and s, and s2
is a cubic or quartic polynomial in t.

Elliptic integrals generally can not be expressed in terms of elementary
functions. However, Legendre showed that all elliptic integrals can be
reduced to the following three canonical forms:

Elliptic Integral of the First Kind (Legendre form)

Elliptic Integral of the Second Kind (Legendre form)

Elliptic Integral of the Third Kind (Legendre form)

where

Note

φ is called the amplitude.

k is called the modulus.

α is called the modular angle.

n is called the characteristic.

Caution

Perhaps more than any other special functions the elliptic integrals
are expressed in a variety of different ways. In particular, the final
parameter k (the modulus) may be expressed using
a modular angle α, or a parameter m. These are related
by:

k = sinα

m = k2 = sin2α

So that the integral of the third kind (for example) may be expressed
as either:

Π(n, φ, k)

Π(n, φ \ α)

Π(n, φ| m)

To further complicate matters, some texts refer to the complement
of the parameter m, or 1 - m, where:

Duplication Theorem

Carlson's Formulas

The conventional methods for computing elliptic integrals are Gauss and
Landen transformations, which converge quadratically and work well for
elliptic integrals of the first and second kinds. Unfortunately they suffer
from loss of significant digits for the third kind. Carlson's algorithm
[Carlson79] [Carlson78],
by contrast, provides a unified method for all three kinds of elliptic
integrals with satisfactory precisions.